JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank Critical Thinking

  • question_answer The angle between the lines joining the points of intersection of line \[y=3x+2\] and the curve \[{{x}^{2}}+2xy+3{{y}^{2}}+4x+8y-11=0\] to the origin, is

    A)            \[{{\tan }^{-1}}\left( \frac{3}{2\sqrt{2}} \right)\]      

    B)            \[{{\tan }^{-1}}\left( \frac{2}{2\sqrt{2}} \right)\]

    C)            \[{{\tan }^{-1}}\left( \sqrt{3} \right)\]                           

    D)            \[{{\tan }^{-1}}\left( \frac{2}{2\sqrt{2}} \right)\]

    Correct Answer: B

    Solution :

               Finding the equation of lines represented by the points of intersection of curve and line with origin, we get \[{{x}^{2}}+2xy+3{{y}^{2}}+(4x+8y)\left( \frac{y-3x}{2} \right)-11{{\left( \frac{y-3x}{2} \right)}^{2}}=0\]            \[\Rightarrow {{x}^{2}}+2xy+3{{y}^{2}}+(2xy-6{{x}^{2}}+4{{y}^{2}}-12xy)\]                                                 \[-\frac{11}{4}{{y}^{2}}-\frac{99}{4}{{x}^{2}}+\frac{33}{2}xy=0\]            Proceed and find the angle between the lines represented by it using\[\alpha ={{\tan }^{-1}}\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}\].

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